Cumulants on the Wiener Space

By Ivan Nourdin, Giovanni Peccati. 2009

71 citations recorded by the source corpus.

We combine infinite-dimensional integration by parts procedures with a recursive relation on moments (reminiscent of a formula by Barbour (1986)), and deduce explicit expressions for cumulants of functionals of a general Gaussian field. These findings yield a compact formula for cumulants on a fixed Wiener chaos, virtually replacing the usual "graph/diagram computations" adopted in most of the probabilistic literature.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8753c8d61dc5a3bf2981b91e4c518f47e305b61c

Citations in the corpus

• The optimal fourth moment theorem (2013, unknown): “ifferentiable and with a bounded derivative, one has that ϕ(F) ∈ D1,2, and moreover Dϕ(F) = ϕ′(F)DF. (2.12) 4 We conclude this section by recalling the definition of the operators Γj, as first defined in [14]; see [15, Chapter 8] for a discussion of recent developments. Definition 2.2 (Gamma operators) L”
• Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants (2011, unknown): “See [12, Theorem 5.1] for details on this point.”
• Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on S^d (2014, unknown): “ alone. Such results are currently being generalized in several directions, including Poisson processes, free probability, random matrices, Markov subordinators and information theory (see [3], [12], [24], [27], [28] e.g.); in the present paper, we will stick to Gaussian subordinated circumstances, ”